Helping a prof with his research vs doing your own research.

So all of my research experience so far has been done with helping a prof with his research. I haven't really done any independent research on my own. Is this the norm among students who apply to grad school? Is it not as good as me doing my own research? (and compared to students who have done their own research projects, is it a significant disadvantage?) Of course I can put it in my personal statement and say that I understood what my role was and the direction the research is supposed to take (e.g. show that I'm perfectly capable of having my own research project should I have the chance to do it)

It's unusual for students who haven't already spent a few years in graduate school to know enough about a topic to come up with something original to study. Grad schools will expect you to have worked with a professor, and you'll be doing that in graduate school as well. Many people finish a PhD without having come up with very original ideas of their own and working on a suggestion from a professor instead.

So all of my research experience so far has been done with helping a prof with his research. I haven't really done any independent research on my own.

Yes. It's normal. Also I don't think any research is truly independent, and if you find yourself a senior professor you'll find yourself talking with other professors and exchanging ideas.

Is it not as good as me doing my own research? (and compared to students who have done their own research projects, is it a significant disadvantage?)

I don't think that an undergraduate who writes a statement of purpose that claims that they have done their own research project without significant assistance and advice from a professor is going to be taken seriously.

Actually all my research paper are written on my own. Although I got some idea on the different references to make my research paper more academic one. Cause if you write your own research paper, you can explain it easily.